Which of the following statements must be true about the polynomial function f(x)?
If 1 + StartRoot 13 EndRoot is a root of f(x), then Negative 1 minus StartRoot 13 EndRoot is also a root of f(x).
If 1 + 13i is a root of f(x), then 1 – 13i is also a root of f(x).
If 13 is a root of f(x), then –13 is also a root of f(x).
If –13 + i is a root of f(x), then 13 + i is also a root of f(x).

Let’s evaluate each statement regarding the polynomial function \( f(x) \) and whether the statements must be true based on properties of polynomials and their roots:

1. **If \( 1 + \sqrt{13} \) is a root of \( f(x) \), then \( -1 - \sqrt{13} \) is also a root of \( f(x) \).**

   - **False**. This statement assumes the function has some symmetry that forces \( -1 - \sqrt{13} \) to also be a root, but there's no general rule that dictates this for real irrational roots unless the polynomial has real coefficients and is structured in a particular way (e.g., if it was quadratic with these roots). Therefore, \( -1 - \sqrt{13} \) does not *have* to be a root.

2. **If \( 1 + 13i \) is a root of \( f(x) \), then \( 1 - 13i \) is also a root of \( f(x) \).**

   - **True**. If a polynomial has real coefficients, complex roots must occur in conjugate pairs. Since \( 1 + 13i \) is a complex number and \( f(x) \) is assumed to have real coefficients, its conjugate \( 1 - 13i \) must also be a root.

3. **If 13 is a root of \( f(x) \), then \( -13 \) is also a root of \( f(x) \).**

   - **False**. This statement suggests that if one root is a positive number, its negative counterpart must also be a root, but this isn't necessarily true unless the polynomial is structured symmetrically (such as an even-degree polynomial). There's no rule that forces \( -13 \) to be a root just because \( 13 \) is.

4. **If \( -13 + i \) is a root of \( f(x) \), then \( 13 + i \) is also a root of \( f(x) \).**

   - **False**. This statement implies that if a complex number with a negative real part is a root, its positive counterpart must also be a root, but complex roots typically occur in conjugate pairs (not in terms of changing the sign of the real part). There's no reason \( 13 + i \) must be a root just because \( -13 + i \) is.

### Summary:
- Statement 2 is **true**.
- Statements 1, 3, and 4 are **false**.

---

Would you like more details on any of these points, or do you have further questions?

### Follow-up Questions:
1. What happens if the polynomial \( f(x) \) has complex coefficients instead of real ones?
2. Can polynomials have roots that are not in conjugate pairs?
3. How do the degrees of polynomials relate to their roots?
4. What are the general rules for finding the roots of a quadratic polynomial?
5. How do you determine the number of real and complex roots a polynomial has?

### Tip:
Remember that polynomials with **real coefficients** always have complex roots in **conjugate pairs**.

Which of the following statements must be true about the polynomial function f(x)? If 1 + √13 is a root of f(x), then -1 - √13 is also a root of f(x). If 1 + 13i is a root of f(x), then 1 – 13i is also a root of f(x). If 13 is a root of f(x), then –13 is also a root of f(x). If –13 + i is a root of f(x), then 13 + i is also a root of f(x).

Explore which statements must be true about polynomial roots, including real and complex roots, and their conjugates. This detailed analysis covers key algebraic concepts such as polynomials, complex numbers, and the fundamental theorem of algebra. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation